FIXED POINTS OF MULTIMAPS WHICH ARE
NOT NECESSARILY NONEXPANSIVE
NASEER SHAHZAD AND AMJAD LONE
Received 16 October 2004 and in revised form 3 December 2004
Let C be a nonempty closed bounded convex subset of a Banach space X whose characteristic of noncompact convexity is less than 1 and T a continuous 1-χ-contractive SL
map (which is not necessarily nonexpansive) from C to KC(X) satisfying an inwardness
condition, where KC(X) is the family of all nonempty compact convex subsets of X. It
is proved that T has a fixed point. Some fixed points results for noncontinuous maps
are also derived as applications. Our result contains, as a special case, a recent result of
Benavides and Ramı́rez (2004).
1. Introduction
During the last four decades, various fixed point results for nonexpansive single-valued
maps have been extended to multimaps, see, for instance, the works of Benavides and
Ramı́rez [2], Kirk and Massa [6], Lami Dozo [7], Lim [8], Markin [10], Xu [12], and
the references therein. Recently, Benavides and Ramı́rez [3] obtained a fixed point theorem for nonexpansive multimaps in a Banach space whose characteristic of noncompact
convexity is less than 1. More precisely, they proved the following theorem.
Theorem 1.1 (see [3]). Let C be a nonempty closed bounded convex subset of a Banach
space X such that α (X) < 1 and T : C → KC(X) a nonexpansive 1-χ-contractive map. If T
satisfies
T(x) ⊂ IC (x)
∀x ∈ C,
(1.1)
then T has a fixed point.
Benavides and Ramı́rez further remarked that the assumption of nonexpansiveness
in the above theorem can not be avoided. In this paper, we prove a fixed point result for
multimaps which are not necessarily nonexpansive. To establish this, we define a new class
of multimaps which includes nonexpansive maps. To show the generality of our result,
we present an example. As consequences of our main result, we also derive some fixed
point theorems for ∗-nonexpansive maps.
Copyright © 2005 Hindawi Publishing Corporation
Fixed Point Theory and Applications 2005:2 (2005) 169–176
DOI: 10.1155/FPTA.2005.169
170
Fixed points of multimaps
2. Preliminaries
Let C be a nonempty closed subset of a Banach space X. Let CB(X) denote the family of all
nonempty closed bounded subsets of X and KC(X) the family of all nonempty compact
convex subsets of X. The Kuratowski and Hausdorff measures of noncompactness of a
nonempty bounded subset of X are, respectively, defined by
α(B) = inf {d > 0 : B can be covered by finitely many sets of diameter ≤ d},
χ(B) = inf {d > 0 : B can be covered by finitely many balls of radius ≤ d}.
(2.1)
Let H be the Hausdorff metric on CB(X) and T : C → CB(X) a map. Then T is called
(1) contraction if there exists a constant k ∈ [0,1) such that
H T(x),T(y) ≤ kx − y ,
∀x, y ∈ C;
(2.2)
∀x, y ∈ C;
(2.3)
(2) nonexpansive if
H T(x),T(y) ≤ x − y ,
(3) φ-condensing (resp., 1-φ-contractive), where φ = α(·) or χ(·) if T(C) is bounded
and, for each bounded subset B of C with φ(B) > 0, the following holds:
φ T(B) < φ(B)
resp., φ T(B) ≤ φ(B) ;
(2.4)
here T(B) = x∈B T(x);
(4) upper semicontinuous if {x ∈ C : T(x) ⊂ V } is open in C whenever V ⊂ X is
open;
(5) lower semicontinuous if the set {x ∈ C : T(x) ∩ V = φ} is open in C whenever
V ⊂ X is open;
(6) continuous (with respect to H) if H(T(xn ),T(x)) → 0 whenever xn → x;
(7) ∗-nonexpansive (see [5]) if for all x, y ∈ C and ux ∈ T(x) with d(x,ux ) = inf {d(x,
z) : z ∈ T(x)}, there exists u y ∈ T(y) with d(y,u y ) = inf {d(y,w) : w ∈ T(y)}
such that
d ux ,u y ≤ d(x, y).
(2.5)
A sequence {xn } is called asymptotically T-regular if d(xn ,Txn ) → 0 as n → ∞.
Let φ = α or χ. The modulus of noncompactness convexity associated to φ is defined
by
∆X,φ () = inf 1 − d(0,A) : A ⊂ BX is convex, with φ(A) ≥ ,
(2.6)
where BX is the unit ball of X. The characteristic of noncompact convexity of X associated
with the measure of noncompactness φ is defined in the following way:
φ (X) = sup ≥ 0 : ∆X,φ () = 0 .
(2.7)
N. Shahzad and A. Lone 171
Note that
∆X,α () ≤ ∆X,χ ()
(2.8)
α (X) ≥ χ (X).
(2.9)
and so
Let C be a nonempty subset of X, Ᏸ a directed set, and {xα : α ∈ Ᏸ} a bounded net in
X. We use r(C, {xα }) and A(C, {xα }) to denote the asymptotic radius and the asymptotic
center of {xα : α ∈ Ᏸ} in C, that is,
r x, xα
= limsup xα − x,
α
= inf r x, xα : x ∈ C ,
A C, xα = x ∈ C : r x, xα = r C, xα .
r C, xα
(2.10)
It is known that A(C, {xα }) is a nonempty weakly compact convex set if C is a nonempty
closed convex subset of a reflexive Banach space. For details, we refer the reader to [1, 3].
Let A be a set and B ⊂ A. A net {xα : α ∈ Ᏸ} in A is eventually in B if there exists α0 ∈ Ᏸ
such that xα ∈ B for all α ≥ α0 . A net {xα : α ∈ Ᏸ} in a set A is called an ultranet if either
{xα : α ∈ Ᏸ} is eventually in B or {xα : α ∈ Ᏸ} is eventually in A − B, for each subset B
of A.
A Banach space X is said to satisfy the nonstrict Opial condition if, whenever a sequence {xn } in X converges weakly to x, then for any y ∈ X,
limsup xn − x ≤ limsup xn − y .
n
(2.11)
n
Let C be a nonempty closed convex subset of a Banach space X and x ∈ X. Then the
inward set IC (x) is defined by
IC (x) = x + λ(y − x) : y ∈ C, λ ≥ 0 .
(2.12)
Note that C ⊂ IC (x) and IC (x) is convex.
We need the following results in the sequel.
Lemma 2.1 (see [9]). Let C be a nonempty closed convex subset of a Banach space X and
T : C → K(X) a contraction. If T satisfies
T(x) ⊂ IC (x)
∀x ∈ C,
(2.13)
then T has a fixed point.
Lemma 2.2 (see [4]). Let C be a nonempty closed bounded convex subset of a Banach space
X and T : C → KC(X) an upper semicontinuous φ-condensing, where φ(·) = α(·) or χ(·).
If T satisfies
T(x) ∩ IC (x) = ∅ ∀x ∈ C,
then T has a fixed point.
(2.14)
172
Fixed points of multimaps
Lemma 2.3 (see [3]). Let C be a nonempty closed convex subset of a reflexive Banach space
X and {xβ : β ∈ D} a bounded ultranet in C. Then
rC A C, xβ
≤ 1 − ∆X,α 1− ) r C, xβ ;
(2.15)
here rC (A(C, {xβ })) = inf {sup{x − y : y ∈ A(C, {xβ })} : x ∈ C }.
3. Main results
Let C be a nonempty weakly compact convex subset of a Banach space X and T : C →
KC(X) a continuous map.
Definition 3.1. The map T is called subsequentially limit-contractive (SL) if for every
asymptotically T-regular sequence {xn } in C,
limsup H T xn ,T(x) ≤ limsup xn − x
n
(3.1)
n
for all x ∈ A(C, {xn }).
Note that if C is a nonempty closed convex subset of a uniformly convex Banach space
and {xn } is bounded, then A(C, {xn }) has a unique asymptotic center, say x0 , and so in
the above definition, we have
limsup H T xn ,T x0
n
≤ limsup xn − x0 .
(3.2)
n
It is clear that every nonexpansive map is an SL map. Several examples of functions can
be constructed which are SL maps but not nonexpansive. We include here the following simple example. We further remark that Theorem 1.1 does not apply to the function
defined below.
Example 3.2. Let C = [0,3/5] with the usual norm and consider the map T(x) = x2 . It is
easy to see that T is an SL map but not nonexpansive. Moreover, T is 1-χ-contractive and
has a fixed point.
We now prove a result which contains Theorem 1.1, as a special case, and is applicable
to the above example.
Theorem 3.3. Let C be a nonempty closed bounded convex subset of a Banach X such that
α (X) < 1 and T : C → KC(X) a continuous, SL, 1-χ-contractive map. If T satisfies
T(x) ⊂ IC (x)
∀x ∈ C,
(3.3)
then T has a fixed point.
Proof. We follow the arguments given in [3]. Let x0 ∈ C be fixed. Define, for each n ≥ 1,
a mapping Tn : C → KC(X) by
1
1
T(x),
Tn (x) := x0 + 1 −
n
n
(3.4)
N. Shahzad and A. Lone 173
where x ∈ C. Then Tn is (1 − 1/n)-χ-contractive. Also Tn (x) ⊂ IC (x) for all x ∈ C. Now
Lemma 2.1 guarantees that each Tn has a fixed point xn ∈ C. As a result, we have
limn→∞ d(xn ,T(xn )) = 0. Let {nα } be an ultranet of the positive integers {n}. Set A =
A(C, {xnα }). We claim that
T(x) ∩ IA (x) = ∅
(3.5)
for all x ∈ A. To prove our claim, let x ∈ A. Since T(xnα ) is compact, we can find ynα ∈
T(xnα ) such that
xn − yn = d xn ,T xn .
α
α
α
α
(3.6)
We also have znα ∈ T(x) such that
yn − zn = d yn ,T(x) .
α
α
α
(3.7)
We can assume that z = limα znα . Clearly, z ∈ T(x). We show that z ∈ IA (x) = {x + λ(y −
x) : λ ≥ 0, y ∈ A}. Since T is an SL map and {xnα } is asymptotically T-regular, it follows
that
limsup H T xnα ,T(x) ≤ limsup xnα − x
α
(3.8)
α
for all x ∈ A. Now
yn − zn = d yn ,T(x)
α
α
α
≤ H T xnα ,T(x)
(3.9)
and so
lim xnα − z = lim ynα − znα α
α
≤ limsup xnα − x = r,
(3.10)
α
where r = r(C, {xnα }). Notice also that z ∈ T(x) ⊂ IC (x) and so z = x + λ(y − x) for some
λ ≥ 0 and y ∈ C. Without loss of generality, we may assume that λ > 1. Now
1
1
y = z+ 1− x
λ
λ
(3.11)
and so
lim xnα − y ≤
α
1
1
lim xnα − x ≤ r.
lim xnα − z + 1 −
λ α
λ α
(3.12)
This implies that y ∈ A and so z ∈ IA (x). This proves our claim. By Lemma 2.3, we have
rC (A) ≤ λr,
(3.13)
174
Fixed points of multimaps
where λ := 1 − ∆X,α (1− ) < 1. Now choose x1 ∈ A and for each µ ∈ (0,1), define a mapping
Tµ : A → KC(X) by
Tµ (x) = µx1 + (1 − µ)T(x).
(3.14)
Then each Tµ is a χ-condensing and satisfies
Tµ (x) ∩ IA (x) = ∅
(3.15)
for all x ∈ A. Now Lemma 2.2 guarantees that Tµ has a fixed point. As a result, we can get
an asymptotically T-regular sequence {xn1 } in A. Proceeding as above, we obtain
T(x) ∩ IA1 (x) = ∅
(3.16)
for all x ∈ A1 := A(C, {xn1α }) and rC (A1 ) ≤ λrC (A). By induction, for each m ≥ 1, we can
find an asymptotically T-regular sequence {xnm }n ⊆ Am−1 . Using the ultranet {xnmα }α , we
construct Am := A(C, {xnmα }) with rC (Am ) ≤ λm rC (A). Choose xm ∈ Am . Then {xm }m is a
Cauchy sequence. Indeed, for each m ≥ 1, we have
xm−1 − xm ≤ xm−1 − xm + xm − xm n
n
≤ diam Am−1 + xnm − xm ,
(3.17)
for all n ≥ 1. Now taking limsup, we see that
xm−1 − xm ≤ diamAm−1 + limsup xm − xm n
m−1
n
≤ 3rC A
(3.18)
≤ 3λm−1 rC (A).
Taking the limit as m → ∞, we get limm→∞ xm−1 − xm = 0. This implies that {xm } is a
Cauchy sequence and so is convergent. Let x = limm→∞ xm . Finally, we show that x is a
fixed point of T. Since T is an SL map, for m ≥ 1, we have
limsup H T xnm ,T xm
n
Now, we have for m ≥ 1,
d xm ,T xm
≤ limsup xnm − xm .
(3.19)
n
≤ xm − xnm + d xnm ,T xnm
m + H T xn ,T xm .
(3.20)
This implies that
d xm ,T xm
≤ 2limsup xm − xnm ≤ 2λ
n
m−1
(3.21)
rC (A).
Taking the limit as m → ∞, we have limm→∞ d(xm ,T(xm )) = 0 and so x ∈ T(x). This com
pletes the proof.
N. Shahzad and A. Lone 175
Theorem 3.3 fails if the assumption that T is an SL map is dropped.
Example 3.4. Let B be the closed unit ball of l2 . Define T : B → B by
T(x) = T x1 ,x2 ,... =
1 − x2 ,x1 ,x2 ,... .
(3.22)
Then T is 1-χ-contractive without a fixed point (see [1, 2]). We can show that this map is
not SL if we consider the following sequence {x(n) } in B:
1
1 1 1
x(1) = 0, √ , √ , √ , √ ,... ,
2 4 8 16
1
1
1
1
1
1
x(2) = 0, √ √ , √ √ , √ √ , √ √ , √ √ , √ √ ,... ,
2 2 2 2 2 4 2 4 2 8 2 8
1
1
1
1
1
1
1
1
1
x(3) = 0, √ √ , √ √ , √ √ , √ √ , √ √ , √ √ , √ √ , √ √ , √ √ ,... ,
3 2 3 2 3 2 3 4 3 4 3 4 3 8 3 8 3 8
(3.23)
and so on.
Corollary 3.5. Let C be a nonempty closed bounded convex subset of a Banach space X
such that α (X) < 1 satisfying the nonstrict Opial condition and T : C → KC(X) a nonexpansive map. If T satisfies
T(x) ⊂ IC (x)
∀x ∈ C,
(3.24)
then T has a fixed point.
Proof. This follows immediately from [2, Theorem 4.5] and Theorem 3.3.
Next we present some fixed point results for ∗-nonexpansive maps.
Theorem 3.6. Let C be a nonempty closed bounded convex subset of a Banach space X such
that α (X) < 1 and T : C → KC(X) a ∗-nonexpansive, 1-χ-contractive map. If T satisfies
T(x) ⊂ IC (x)
∀x ∈ C,
(3.25)
then T has a fixed point.
Proof. Define
PT (x) = ux ∈ T(x) : d x,ux = d x,T(x)
(3.26)
for x ∈ C. Since T(x) is compact, PT (x) is nonempty for each x. Furthermore, PT is
convex and compact valued since T is so. Also, PT is nonexpansive because T is ∗nonexpansive. Let B be a bounded subset of C. Then it is easy to see that PT (C) is a
bounded set and χ(PT (B)) ≤ χ(B). Thus PT is 1-χ-contractive. PT also satisfies
PT (x) ⊂ IC (x)
∀x ∈ C.
Now Theorem 3.3 guarantees that PT has a fixed point. Hence T has a fixed point.
(3.27)
176
Fixed points of multimaps
Similarly, we get the following corollary, which extends [11, Theorem 2] to non-selfmultimaps and to spaces satisfying the nonstrict Opial condition.
Corollary 3.7. Let C be a nonempty closed bounded convex subset of a Banach space
X such that α (X) < 1 satisfying the nonstrict Opial condition and T : C → KC(X) a ∗nonexpansive map. If T satisfies
T(x) ⊂ IC (x)
∀x ∈ C,
(3.28)
then T has a fixed point.
Acknowledgment
The authors are indebted to the referees for their valuable comments.
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
J. M. Ayerbe Toledano, T. D. Benavides, and G. López Acedo, Measures of Noncompactness in Metric Fixed Point Theory, Operator Theory: Advances and Applications, vol. 99,
Birkhäuser, Basel, 1997.
T. D. Benavides and P. L. Ramı́rez, Fixed-point theorems for multivalued non-expansive mappings
without uniform convexity, Abstr. Appl. Anal. (2003), no. 6, 375–386.
, Fixed point theorems for multivalued nonexpansive mappings satisfying inwardness conditions, J. Math. Anal. Appl. 291 (2004), no. 1, 100–108.
K. Deimling, Multivalued Differential Equations, de Gruyter Series in Nonlinear Analysis and
Applications, vol. 1, Walter de Gruyter, Berlin, 1992.
T. Husain and A. Latif, Fixed points of multivalued nonexpansive maps, Math. Japon. 33 (1988),
no. 3, 385–391.
W. A. Kirk and S. Massa, Remarks on asymptotic and Chebyshev centers, Houston J. Math. 16
(1990), no. 3, 357–364.
E. Lami Dozo, Multivalued nonexpansive mappings and Opial’s condition, Proc. Amer. Math.
Soc. 38 (1973), 286–292.
T. C. Lim, A fixed point theorem for multivalued nonexpansive mappings in a uniformly convex
Banach space, Bull. Amer. Math. Soc. 80 (1974), 1123–1126.
, A fixed point theorem for weakly inward multivalued contractions, J. Math. Anal. Appl.
247 (2000), no. 1, 323–327.
J. T. Markin, A fixed point theorem for set valued mappings, Bull. Amer. Math. Soc. 74 (1968),
639–640.
H. K. Xu, On weakly nonexpansive and ∗-nonexpansive multivalued mappings, Math. Japon. 36
(1991), no. 3, 441–445.
, Multivalued nonexpansive mappings in Banach spaces, Nonlinear Anal. Ser. A: Theory
Methods 43 (2001), no. 6, 693–706.
Naseer Shahzad: Department of Mathematics, Faculty of Sciences, King Abdul Aziz University,
P.O. Box 80203, Jeddah 21589, Saudi Arabia
E-mail address: nshahzad@kau.edu.sa
Amjad Lone: Department of Mathematics, College of Sciences, King Khalid University, P.O. Box
9004, Abha, Saudi Arabia
E-mail address: amlone@kku.edu.sa